Chapter 1 - affine geometry

Abstract

This chapter focuses on the study of affine geometry. Affine geometry is what remains after practically all ability to measure length, area, and angles, has been removed from Euclidean geometry. The whole theory of homothetic figures lies within affine geometry. The notions of translation and magnification—these are the dilations—are in the domain of affine geometry and, more generally, affine transformations—one-to-one, onto functions which preserve parallelism—constitute an affine notion. The chapter discusses axioms for affine geometry. A division ring for which multiplication is commutative is called a field. A model for n-dimensional affine space is reviewed. The affine sub-spaces of dimension n -1 are called hyperplanes. As the intersection of two affine subspaces can be empty and an affine subspace is never empty, the intersection of two affine subspaces is not always an affine subspace. With the exception of the empty intersection, the intersection of affine subspaces is an affine subspace.